Global Dynamics of a Mathematical Model on Smoking

Applicationes Mathematicae, 2017

In this work, we derive and analyze a smoking model by taking into account hospitalized smokers and smoke quitters of two types: temporary and permanent, which is one of the possible extensions of the giving-up smoking model. Temporary quitters may become smokers again, while permanent quitters, once they quit, never smoke again in the entire life span. The existence and stability of the possible equilibria of the model are examined in terms of a certain threshold condition R, the smoking generation number (the basic reproduction number). Numerical simulations are carried out to investigate the influence of the key parameters on the spread of smoking, to support the analytical conclusion and illustrate possible behavioral scenarios. 1. Introduction. Mathematical models can provide a useful tool to analyze the spread and control of diseases [2, 4, 13, 15]. Castillo-Garsow et al. [5] developed a mathematical model of giving-up smoking and analyzed the dynamical behavior of different smoking individuals. Hoogenveen et al. [10] presented a model that was capable of describing the effects of smoking cessation on morbidity and mortality over time. In their model they described the life course of quitters after smoking cessation, taking into account relapse. The basic model of [5] was extended by Sharomi and Gumel [14] to account for variability in smoking frequency, by introducing the classes of mild and of chain smokers as well as the develop

Applied Mathematics and Computation, 2008

This paper provides a rigorous mathematical study for assessing the dynamics of smoking and its public health impact in a community. A basic mathematical model, which is a slight refinement of the model presented in [F. Brauer, C. Castillo-Chavez. Mathematical models for the dynamics of tobacco use, recovery and relapse. Technical Report Series, BU-1505-M. Department of Biometrics, Cornell University. 2000], is designed first of all. It is based on subdividing the total population in the community into non-smokers, smokers and those smokers who quit smoking either temporarily or permanently. The theoretical analysis of the basic model reveals that the associated smoking-free equilibrium is globally-asymptotically stable whenever a certain threshold, known as the smokers-generation number, is less than unity, and unstable if this threshold is greater than unity. The public health implication of this result is that the number of smokers in the community will be effectively controlled (or eliminated) at steady-state if the threshold is made to be less than unity. Such a control is not feasible if the threshold exceeds unity (a global stability result for the smoking-present equilibrium is provided for a special case). The basic model is extended to account for variability in smoking frequency, by introducing two classes of mild and chain smokers as well as the development and the public health impact of smoking-related illnesses. The analysis and simulations of the extended model, using an arbitrary but reasonable set of parameter values, reveal that the number of smokers in the community will be significantly reduced (or eliminated) if chain smokers do not remain as chain smokers for longer than 1.5 years before reverting to the mild smoking class, regardless of the time spent by mild smokers in their (mild smoking) class. Similarly, if mild smokers practice their mild smoking habit for less than 1.5 years, the number of smokers in the community will be effectively controlled irrespective of the dynamics in the chain smoking class.

Advances in Difference Equations, 2016

We study the qualitative behavior of a smoking model in which the population is divided into five classes, that is, non-smokers, smokers, people who temporarily quit smoking, people who permanently quit smoking, and people who are associated with illness due to smoking. The global asymptotic stability of the unique positive equilibrium point is presented. More precisely, a graph-theoretic method is used to prove the global stability of the unique positive equilibrium point.

International Journal of Applied Mathematical Research, 2014

In this paper we present and analyze a generalization of the giving up smoking model that was introduced by Sharomi and Gumel [4], in which quitting smoking can be temporary or permanent. In our model, we study a population with peer pressure effect on temporary quitters and we consider also the possibility of temporary quitters becoming permanent quitters and the impact of this transformation on the existence and stability of equilibrium points. Numerical results are given to support the results.

International Journal of Differential Equations, 2021

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, ...

Nonlinear Engineering

The square-root dynamics of smoking model with cravings to smoke, in which square root of potential smokers and smokers is the interaction term, has been studied. We categorized net population in four different chambers: non-smokers/potential smokers, smokers/infected people, non-permanent smokers/temporary quitters and the permanent quitters. By dynamical systems approach, we analyzed our model. Moreover, for proving the unique equilibrium point to be globally stable, we took help of graph theoretic approach. The sensitivity analysis of the model is performed through the diseased classes effectively to design reliable, robust and stable control strategies. The model is designed like optimal control trouble to find out importance of various control actions on our system that are insisted by the sensitivity analysis. We have applied two controls, which are the awareness campaign through the media transmission to control the potential smokers and temporary quit smokers to become smoke...

2011

Smoking is a large problem in the entire world. Despite overwhelm- ing facts about the risks, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The main purpose of this paper is to determine the asymptotic behavior of a math- ematical model using for giving up smoking. Our interest here is to derive and analysis the model taking into account the occasional smokers compartment in the giving up smoking model. Analysis of this model reveals that there are four equilibria, one of them is the smoking-free and the other three correspond to presence of smoking. We also present the global stability and parameter estimates that characterize the natural history of this disease with numerical simulations.

Contemporary Engineering Sciences, 2019

This study presents a mathematical model that represents the population growth dynamics of tobacco consumers based on a system of ordinary nonlinear differential equations. The model is used to determine the Basic Reproductive Number (R 0). Two points of equilibrium are found and their local stability is classified. Finally, the Matlab software is used to present numerical simulations using the fourth-order Runge-Kutta method, and it is shown that the solutions approach an asymptotically stable point, under the variation of R 0 .

Our aim in this paper, is first constructing a Lyapunov function to prove the global stability of the unique smoking-present equilibrium state of a mathematical model of smoking. Next we incorporate random noise into the deterministic model. We show that the stochastic model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. Then a stochastic Lyapunov method is performed to obtain the sufficient conditions for mean square and asymptotic stability in probability of the stochastic model. Our analysis reveals that the stochastic stability of the smoking-present equilibrium state, depends on the magnitude of the intensities of noise as well as the parameters involved within the model system.

The Bulletin of the Malaysian Mathematical Society Series 2

Smoking is a large problem in the entire world. Despite overwhelming facts about the risks, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The main purpose of this paper is to determine the asymptotic behavior of a mathematical model using for giving up smoking. Our interest here is to derive and analysis the model taking into account the occasional smokers compartment in the giving up smoking model. Analysis of this model reveals that there are four equilibria, one of them is the smoking-free and the other three correspond to presence of smoking. We also present the global stability and parameter estimates that characterize the natural history of this disease with numerical simulations.

American Journal of Computational Mathematics

In this paper, the global stability of free smoking equilibrium point was evaluated and presented graphically. The linear stability of a developed mathematical model illustrates the effect on the population of chain, mild and passive smokers. MATLAB programming was used to simulate the solutions, the reproduction number 0 R and the nature of the equilibria.

Mediterranean Journal of Mathematics, 2018

Smoking impacts health and as a result creates several problems related to age which means smoking has a strong correlation with age. Keeping this problem in view, we consider the global asymptotic properties of age-structured smoking model. First, we formulate the model and present the existence and uniqueness of solution. Then we discuss the equilibrium points and construct the Lyapunov function to examine global stability of the free smoking and positive smoking equilibrium points. Finally, we fixed the age factor and use the non-standard finite difference (NSFD) scheme for numerical solutions and compare our results obtained with RK4 and ODE45 graphically with the help of MATLAB.

AIP Advances, 2018

In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.

Boundary Value Problems, 2020

This research work is related to a tobacco smoking model having a significance class of users of tobacco in the form of snuffing. For this purpose, the formulation of the model containing snuffing class is presented; then the equilibrium points as regards being smoking free and smoking positive are discussed. The Hurwitz theorem is used for finding the local stability of the model and Lyaponov function theory is used for the search of global stability. We use different controls for control of smoking and the Pontryagin maximum principle for characterization of the optimal level. For the solution of the proposed model, a nonstandard finite difference (NSFD) scheme and the Runge–Kutta fourth order method are used. Finally, some numerical results are presented for control and without control systems with the help of MATLAB.

Journal of the National Science Foundation of Sri Lanka, 2021

In this paper, the deterministic model of smoking consisting of five classes, has been qualitatively analysed. Explicit formula for the reproduction number has been obtained. Equilibria have been found and their global asymptotic stability has been discussed. Method of matrix theoretic, with the Perron eigenvectors, is used to get the global asymptotic stability of smoking-free equilibrium. It is shown that unique endemic equilibrium is globally asymptotically stable by using graph theoretic approach. To know the important factors, through which the disease spreads rapidly, sensitivity analysis of basic reproduction number and endemic level of smokers has been performed. This sensitivity analysis urged to modify the existing problem by inserting two controls namely prevention and treatment. Optimal control problem has been designed on the basis of sensitivity analysis. The existence of controls has been proved analytically and numerically it is shown that these applied controls sign...